Synthetic Division
When dividing a polynomial by a binomial of the form x c, we can use synthetic
division to speed up the process. For synthetic division we write only the essential
parts of ordinary division. For example, to divide x^{3}  5x^{2}
+ 4x  3 by x  2, we write only the coefficients of the dividend 1, 5, 4, and
3 in order of
descending exponents. From the divisor x  2 we use 2 and start with the following
arrangement:
2 
1  5 4  3 
(1 Â· x^{3}  5x^{2} + 4x  3) Ã· (x  2) 
Next we bring the first coefficient, 1, straight down:
We then multiply the 1 by the 2 from the divisor, place the answer under the
5, and
then add that column. Using 2 for x  2 allows us to add the column rather than
subtract as in ordinary division:
We then repeat the multiplyandadd step for each of the remaining columns:
From the bottom row we can read the quotient and remainder. Since the degree of
the quotient is one less than the degree of the dividend, the quotient is 1x^{2}
 3x  2. The remainder is 7.
The strategy for getting the quotient Q(x) and remainder R by synthetic
division can be stated as follows.
Strategy for Using Synthetic Division
1. List the coefficients of the polynomial (the dividend).
2. Be sure to include zeros for any missing terms in the dividend.
3. For dividing by x c, place c to the left.
4. Bring the first coefficient down.
5. Multiply by c and add for each column.
6. Read Q(x) and R from the bottom row.
Caution
Synthetic division is used only for dividing a polynomial by
the binomial x  c, where c is a constant. If the binomial is x  7, then c
= 7. For
the binomial x + 7 we have x + 7 = x  ( 7) and c =  7.
Example
Using synthetic division
Find the quotient and remainder when 2x^{4}  5x^{2} + 6x  9 is divided by x
+ 2.
Solution
Since x + 2 = x  (2), we use 2 for the divisor. Because x^{3} is missing in the
dividend, use a zero for the coefficient of x^{3}:
Because the degree of the dividend is 4, the degree of the quotient is 3. The quotient
is 2x^{3}  4x^{2} + 3x, and the remainder is 9. We can also express the results of this
division in the form quotient
:
2x^{4}  5x^{2 }+ 6x  9 
= 2x^{3}  4x^{2} + 3x + 
9 
x + 2 

x + 2 